Consider the following statement which is a part of Conjecture 1.3 in the paper titled "The asymptotic growth of torsion homology for arithmetic groups" authored by N. Bergeron and A. Venkatesh.
Let $G$ be a $\mathbb{Q}$-semisimple group of zero defect(includes the Shimura variety case) and let $K \subset G(\mathbb{R})$ be a maximal compact subgroup and $\Gamma \subset G(\mathbb{Q})$ be a congruence subgroup. Denote the locally symmetric space $\Gamma \backslash G(\mathbb{R})/K$ by $X_{\Gamma}$. Let $\Gamma_N$ denote the descending family of congruence subgroups such that $\cap \Gamma_N = \lbrace 1 \rbrace$. Then $$ \lim_{N \to \infty} \frac{\log |\mathrm{H}^i(X_{\Gamma_N}, \mathbb{Z})_{tors}|}{[\Gamma : \Gamma_N]} = 0. $$
I was wondering what are heuristics for this conjecture? What are the evidence available for this conjecture? Is something known in the case of Shimura varieties? Any help of reference is appreciated. Thank you.